Download General Pontryagin-Type Stochastic Maximum Principle and by Qi Lü, Xu Zhang (auth.) PDF

By Qi Lü, Xu Zhang (auth.)

The classical Pontryagin greatest precept (addressed to deterministic finite dimensional keep watch over platforms) is among the 3 milestones in smooth keep watch over concept. The corresponding thought is by means of now well-developed within the deterministic endless dimensional surroundings and for the stochastic differential equations. besides the fact that, little or no is understood concerning the comparable challenge yet for managed stochastic (infinite dimensional) evolution equations while the diffusion time period includes the keep watch over variables and the keep an eye on domain names are allowed to be non-convex. certainly, it really is one of many longstanding unsolved difficulties in stochastic keep watch over concept to set up the Pontryagin style greatest precept for this type of basic regulate platforms: this publication goals to offer an answer to this challenge. This ebook can be worthwhile for either rookies and specialists who're drawn to optimum keep watch over conception for stochastic evolution equations.

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Extra resources for General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions

Example text

K∀≥ Moreover, ||G ||L (X,L q1 (0,T ;L q2 (Ω;Y ))) ≤ supn∞N ||Gn ||L (X, L q1 (0,T ;L q2 (Ω;Y )) . 1 (Hence, the detailed proof will be omitted). 2 Let X and Y be accordingly a separable and a reflexive Banach space, and let L p (Ω, FT , P), with 1 ≤ p < ≥, be separable. Let 1 ≤ p1 < ≥ and is a sequence of uniformly bounded, pointwisely 1 < q1 < ≥. Assume that {Gn }≥ n=1 p q defined linear operators from L F1T (Ω; X ) to L F1 T (Ω; Y ). Then, there exist a subp1 q1 ≥ sequence {Gn k }≥ k=1 ∈ {Gn }n=1 and an G ∞ L pd L FT (Ω; X ), L FT (Ω; Y ) such that q p G u(·) = (w) − lim Gn k u(·) in L F1 T (Ω; Y ), ∗ u(·) ∞ L F1T (Ω; X ).

Let 1 ≤ p1 < ≥, 1 < q1 , q2 < ≥ and 0 ≤ t0 ≤ T . Assume that {Gn }≥ n=1 isp a sequence of uniformly bounded, pointwisely defined linear operators from L F1t (Ω; X ) to 0 ≥ L F1 (t0 , T ; L q2 (Ω; Y )). Then, there exist a subsequence {Gn k }≥ k=1 ∈ {Gn }n=1 and an q p q G ∞ L pd L F1t (Ω; X ), L F1 (t0 , T ; L q2 (Ω; Y )) 0 such that q p G u(·) = (w) − lim Gn k u(·) in L F1 (t0 , T ; L q2 (Ω; Y )), ∗ u(·) ∞ L F1t (Ω; X ). 0 k∀≥ Moreover, ||G ||L (L p1 Ft0 (Ω;X ), q L F1 (t0 ,T ;L q2 (Ω;Y ))) ≤ sup ||Gn ||L (L p1 Ft0 (Ω;X ), n∞N L F1 (t0 ,T ;L q2 (Ω;Y )) .

Step 2. Denote by A the infinitesimal generator of {T (t)}t∗0 . 19) f (t, P, Q) = −J → P − PJ − K → PK − K → Q − QK + F. 9). 1, where the Hilbert space H is replaced by L2 (H). 15). 14). e. (t, ω) ≥ [0, T ] × Ω and x ≥ H. Hence, O(t, ω) ≥ L2 (H). For any λ ≥ ρ(A), define a family of operators {Tλ (t)}t∗0 on L2 (H) as follows: Tλ (t)O = Sλ (t)OSλ→ (t), ∞ O ≥ L2 (H). 4 Well-Posedness Result for the Operator-Valued BSEEs 43 By the result proved in Step 1, it follows that {Tλ (t)}t∗0 is a C0 -semigroup on L2 (H).

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